From discrete time series models to continuous

Question

Is it possible to convert an SARIMA model to a continuous model?

If so, what is the methodology to do that?

Answer 1

Let's look at the formula for an ARMA(p, q) model

$$ X_t = c + \sigma z_t + \sum_{i=1}^p \varphi_i X_{t-i} + \sigma\sum_{i=1}^{q}\theta_i z_{t-i} $$

where $z_t \in \mathcal{N}(0,1)$ for all $t$. Transform this into the continuous time counterpart below.

$$ X_t = c + \sigma W_t + \int_0^{T_p}\varphi(v)X_{t-v} dv + \sigma\int_0^{T_q}\theta(v)W_{t-v}dv $$

Where $T_{p,q}$ are the width of the lookback window (in years) for the AR and MA terms. The sequence of coefficients have been replaced by continuous functions of time $\varphi, \theta : \mathbb{R_+} \mapsto \mathbb{R}$. Interpolation methods can be used to go from the discrete $\varphi_i$ to the continuous $\varphi(v)$.

A problem that I see with this is that I believe the $z_t$ should probably not be replaced with $W_t$ but something more like "$\sqrt{dt}dW_t$ " but I am not sure how to formalize that. I am also not sure about how to write out the total derivative for this SDE $dX_t$ any help with that would be appreciated.

Answer 2

You find the connection between an AR(1) and an Ornstein-Uhlenbeck process here in QSE if you search.

Taken the description form here (which is just the way to do it):

$$ X_{n+1} = c + a X_n + b \varepsilon_k $$ and by setting $c=\theta \mu \Delta t, a=−\theta \Delta t$ and $b =\sigma \sqrt{\Delta t}$ you will get the discrete time approximation $$ X_{n+1} = \theta(\mu - X_n)\Delta t + \sigma \varepsilon_k\sqrt{\Delta t}, $$ which corresponds to the continuous time OU-process.

If you add a seasonal process then you have SAR(1).

Thus AR(1) + seasonal component in discrete time is $$ dX_t = a (b-\mu) dt + dB_t + dS_t $$ where $S_t$ is a periodic function. This would be the solution for the AR(1) case and no seasonal AR or MA component.

If we think of the dependence structure in higher order AR processes I wonder whether there is a continuous time model for AR(n) for $n>1$.